CHERYL'S BIRTHDAY - THE FINAL VERDICT

It must be a sign of the times that a simple maths question from Singapore can become an overnight internet sensation last week. Gone were the times when those equations and formulas simply evaporated and wafted away as soon as you walk out of the exam hall. No way. Now you see and hear debate and discussion everywhere between the majority left-brain 'July 16' people and a smaller yet belligerent, supposedly right-brain 'August 17' faction who feel obligated to rebel against the rather totalitarian-sounding Official Correct Answer.

Just as quickly as when it first exploded, the fever has almost completely subsided now save for a few pockets of die-hard people still suffering from the withdrawal symptoms of their once-in-a-lifetime intellectual exertion. You can easily identify them when you find them suddenly stopping at a street corner with their jaw tilted 30 degrees seemingly deep in thought. I'm writing this article to help put these people out of their misery and hopefully coax them back to become productive members of the economy again.

Preamble

Because of the high stakes involved, secondary school maths is no longer considered adequate to solve this. So a wider array of methods including common sense, logic, philosophy, critical thinking, linguistic analysis and other uses of human reason are employed.

The two men wanted to know when Cheryl's birthday is (never mind what they planned to do!) and apparently were engaged in a dialogue to help each other find the answer without saying aloud what Cheryl had already told them separately. All ten dates as given by Cheryl were possible.

All details of the narrative as set by SASMO will be taken literally.

We must remember that we should take the point of view of either Albert or Bernard when trying to analyze their actions. We must not take the audience's perspective because each of them had different priorities and had different sets of information available to them.

Though this is supposed to be a maths problem, it is only mathematical for the audience. Albert and Bernard were not taking a maths test - they were trying to solve a puzzle and they should be allowed to behave like normal human beings, thinking and acting logically but occasionally prone to human failings and limitations.

Let's just restrict our considerations to the 2 possible answers, viz. July 16 and August 17, ie, Albert was told either July or August and Bernard 16 or 17.

It is my opinion that the common method of reasoning from the audience's point of view and eliminating invalid dates as we follow the dialogue is actually wrong. I believe that we should reverse the process. Start at the end and work backwards from the 2 scenarios, each bearing an assumed answer and see whether they satisfy the narrative and the behaviour of the 2 men. The scenario that is the more plausible and has fewer contradictions will be the correct one.

3 areas of ambiguity:

(1) Albert said: "I don't know when Cheryl's birthday is, but I know that Bernard does not know too."

Let's break what he said into 2 parts. The first part is "I don't know when Cheryl's birthday is, ...". Here, either he was just declaring that he was starting the game with no knowledge or he was trying to tell Bernard that the month is not June (June 17 becomes a giveaway date when June 18 has been eliminated as one of the initial giveaway dates) .

But, he went on to say that "... but I know that Bernard does not know too." Was he saying that to express his thought that they were starting the game with equal ignorance? Or was he trying to tell Bernard that he was told July or August, and therefore he knew that Bernard could not be told 18 or 19 (being unique dates, June 18 and May 19 are giveaway dates). Was he trying to help Bernard rule out the months of May and June? Was he successful in directing Bernard to think in this way?

The first ambiguity of superfluity arose when we realize that if saying "..but I know that Bernard does not know too" already excludes May and June, why also say "I don't know when Cheryl's birthday is, ..." just to exclude June? Furthermore, the half-statement to exclude June came before the half-statement that excludes May and June. If both halves of the statement are clues to Bernard, then the first half "I don't know when Cheryl's birthday is, ..." is superfluous! Albert could have just said "I'm certain that Bernard does not know Cheryl's birthday now." That would have no ambiguity and made Bernard sit up and realize that it was an important deliberate clue.

But, in the sequence that it came, "exclude June!" hit Bernard first and the significance of the second part might escape Bernard or it might lead Bernard to think that the second part was not intended to be a clue. There is something wrong here. Now, I'm not sure whether Albert intended only the first half to be a clue, only the second half to be a clue, both are intended clues or perhaps, the whole statement contains no clue(s) at all!

(2) Bernard replied: "At first I don't know when Cheryl's birthday is, but now I know."

Again, we break his reply into 2 parts. The first part "At first I don't know when Cheryl's birthday is, .." seems to be a statement of fact in reply to affirm the second half of Albert's statement. If that was the case, Bernard had probably taken the second half of Albert's statement not as a clue but as a correct statement of his own position. This first part "At first I don't know when Cheryl's birthday is,.." seems superfluous as Albert already said he knew that Bernard didn't know, but if Bernard found it necessary to say it, it suggests that he might have thought that Albert was merely guessing.

The second part: ".. but now I know" suggested that something Albert said gave him the answer.

Now, a second ambiguity of attribution arises: which part of Albert's statement gave him his answer?

Imagine, you were Bernard and you were told 17 and the dialogue had not started yet; what would you be doing? You would be looking at June 17 and August 17 only and you would not be too bothered by the other dates. You knew that if Albert knew that you were not told 18 (which you weren't), his declaration of not having the answer in the first instance would be the very thing you need to know, to exclude June 17 and clinch the August 17 answer. You could announce your triumphant success in a flash because that was precisely what you were waiting for. If you look at the timing of the dialogue, it seems to support this theory.

In a normal conversation, it takes about 5 seconds (yes, I timed my wife saying the statement at a normal pace) for Albert to say "I don't know when Cheryl's birthday is, but I know that Bernard does not know too." By mid-sentence, Bernard would have heard what he wanted to hear and 2.5 seconds later, Albert finished his statement. Bernard would take a 3-second pause to start his reply. That is to say, he must have decided that he got the answer in 3 seconds flat or less. Bernard could not do any thinking during the 2.5 seconds when Albert was saying his second half as he was listening to it and he could not do any thinking when he was delivering his reply. So, Bernard's quick response in announcing his success suggests that August 17 is the answer.

On the other hand, if you were Bernard and was told 16, you would be deciding between May 16 and July 16. Before the dialogue started, you would be wondering as to what Albert could say to help you arrive at the answer. You would be less prepared than if you were told 17. So, when Albert finished his sentence, you could only register that June is definitely out, but you would be a little surprised and caught off-guard that Albert said that he knew that you didn't have the answer yet. Your first thought would be: "wow, how come he knew?", "has he been told by someone?", "can he read my mind?", and then later: "what's the significance of that?" and then many seconds later, if not minutes, begin to see the light: "could it be that Albert was being told July or August, and he was telling me that May and June are out, so the answer should be July 16!" Because Bernard was probably unprepared for what Albert said in the second half of this statement, he could not do all that impromptu thinking within the 3 seconds as suggested in the dialogue. His arrival at the answer so soon after the commencement of the dialogue seems to be against July 16 as the answer.

(3) Albert followed up: "Then I also know when Cheryl's birthday is."

Now, the third ambiguity of mind-reading concerns whether Albert read Bernard's mind correctly or not. Albert must judge: "did Bernard rule out May or not?". If Albert was optimistic, he would assume that Bernard had got all his clues (including the superfluous one) within 3 seconds and would interpret them correctly by excluding both May and June. He would have thought that Bernard is a very fast thinker, was prepared for what he said or could probably read Albert's mind. So, he proceeded to do his deductions, but if Bernard did not think or act like he assumed (ie in excluding May), we don't know what answer he would get.

If he was a little bit more careful, he would reflect: "I hope this guy got my clue, though I messed it up a bit", "well, I could rule out May and June because Cheryl told me it was July or August", "but, this guy (Bernard) wasn't even told about months at all and just getting my clue (assuming he got the right one) may or may not make him rule out May", "why did he restate that he didn't know initially when I already told him I knew that?", "could Bernard be having doubts about what I'm trying to tell him about excluding May and June?, "well, ok, let's say he retained May in his deliberations", "well, then there is only a single 17 in his remaining 7 options and this guy responded so fast, then it must be August 17!" (since Cheryl had told me August).

Cheryl's Birthday is Quantum Physics

So now we can conclude.

If the answer is July 16, Bernard must have been very alert and had superfast mental reflexes, was wise enough to discard Albert's superfluous clue, was not surprised by Albert's proclaimed knowledge of his own initial ignorance, did not doubt it despite finding it necessary to restate his ignorance, quickly excluded May and June in his calculations, and having read Albert's mind perfectly, came to a quick conclusion. Albert would have been glad that Bernard thought like him and confirmed his answer.

If the answer is August 17, Albert's claim that "but I know that Bernard does not know too" did not strike Bernard as a clue to exclude May and June (again) because 2.5 seconds earlier, Albert had just told him a clue to exclude June. Excluding June is already enough for him to come to a quick conclusion. The game ended almost immediately after it started (after one sentence). Albert realized that, due his own fault, his second clue to exclude both May and June might not have got through to Bernard. Alternatively, his statement "but I know that Bernard does not know too" was made for reasons other than as a clue. He tried to think like Bernard under those circumstances and arrived at the answer.

There is no unequivocal answer. The answer can change depending on which way the above 3 ambiguities swing (or rather your own preferred way?). Like Heisenberg's Uncertainty Principle, the answer changes as your thinking about it changes. The answer is not determined by Cheryl, Albert nor Bernard but by our judgement about what Albert surmised to be Bernard's thinking and his theoretical choices when he declared that he found the answer. What Bernard actually found became unimportant. The answer is not already there to compel us to acknowledge. It is our interpretation of the interactive thinking and actions of the two men that makes us waver between one answer and the other.

My feeling is that if plausibility and correspondence with the narrative, the natural progression of the dialogue, the time-frame in which everything occurred (the game was over after the first sentence of the dialogue),  August 17 would get my vote.